We report the detection of extended X-ray emission around two powerful radio galaxies at z approx. 3.6 (4C 03.24 and 4C 19.71) and use these to investigate the origin of extended, inverse Compton (IC) powered X-ray halos at high redshifts. The halos have X-ray luminosities of L(sub X) approx. 3 x 10(exp 44) erg/s and sizes of approx.60 kpc. Their morphologies are broadly similar to the approx.60 kpc long radio lobes around these galaxies suggesting they are formed from IC scattering by relativistic electrons in the radio lobes, of either cosmic microwave background (CMB) photons or far-infrared photons from the dust-obscured starbursts in these galaxies. These observations double the number of z > 3 radio galaxies with X-ray-detected IC halos. We compare the IC X-ray-to-radio luminosity ratios for the two new detections to the two previously detected z approx. 3.8 radio galaxies. Given the similar redshifts, we would expect comparable X-ray IC luminosities if millimeter photons from the CMB are the dominant seed field for the IC emission (assuming all four galaxies have similar ages and jet powers). Instead we see that the two z approx. 3.6 radio galaxies, which are 4 fainter in the far-infrared than those at z 3.8, also have approx.4x fainter X-ray IC emission. Including data for a further six z > or approx. 2 radio sources with detected IC X-ray halos from the literature, we suggest that in the more compact, majority of radio sources, those with lobe sizes < or approx.100-200 kpc, the bulk of the IC emission may be driven by scattering of locally produced far-infrared photons from luminous, dust-obscured starbursts within these galaxies, rather than millimeter photons from the CMB. The resulting X-ray emission appears sufficient to ionize the gas on approx.100-200 kpc scales around these systems and thus helps form the extended, kinematically quiescent Ly(alpha) emission line halos found around some of these systems. The starburst and active galactic nucleus activity in these galaxies are thus combining to produce an even more effective and widespread "feedback" process, acting on the long-term gas reservoir for the galaxy, than either individually could achieve. If episodic radio activity and co-eval starbursts are common in massive, high-redshift galaxies, then this IC-feedback mechanism may play a role in affecting the star formation histories of the most massive galaxies at the present day.